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Internat. J. Math. & Math. Sci.VOL. 15 NO. 4 (1992) 697-700

697

AN INFINITE VERSION OF THE PLYA ENUMERATION THEOREM

ROBERT A. BEKES

Mathematics Department, Santa Clara UniversitySanta Clara, CA 95053

(Received July 29, 1991 and in revised form December 17, 1991)

ABSTRACT. Using measure theory, the orbit counting form of P61ya’s enumeration theorem

is extended to countably infinite discrete groups.

KEY WORDS AND PHRASES. Infinite discrete group, P61ya’s emumeration theorem.

1991 AMS SUBJECT CLASSIFICATION CODE. 05A15

1. INTRODUCTION.Let G be countable discrete group acting as permutations on a countable set D. Let S be a

nite set with cardinality, SI N. Denote by S the set of functions from D to S. For 3’ Sdefine g S by g(d)=7(9ld). For a subgroup K of G let mK be a set of representatives for

the orbits of K in S. Let y be a Hilbert space with orthonormal basis {%,: 3" ’5} and inner

product < >. Define a unitary representation of G on by ,t(9)e3"= %3".The number of orbits of G in SDis denoted by AGI. For finite G and D this can be

counted by the P61ya enumeration theorem. Specifically, for each 9 G, let ci(g)be the

number of cycles of length in the representation of 9 as a product of disjoint cycles in D and

let M(g)=vlc(a)...V,,c’(a), where n=IDI. The cycle index of G on D is the polynomial PG=1 M(g). Denote by aPG the value Pa at y,=N, i=1 to n. P61ya’s enumeration theorem,G,g G

see P61ya [1], says that IAal aPa.Define the operator TG on X by Ta= ,(g). Then it can also be shown, see

g GWilliamson [2], that IAGI- trace(Ta on ). It is these two ways of measuring a set of

representatives for orbits that we extend to infinite G and D.

2. THE MAIN RESULTS.If we view S as a finite group with the discrete topology, then SD is a compact group in the

product topology. Let be normalized Haar measure on S.={I if g3’=3’For g G and 3’ sD define f(3")= < r (g)e3", e3’ >. Then f(7) 0 otherwise.

LEMMA 1. f is measurable.

7(1d=7(d)PROOF. Let f.[d,,=’_oherwise

and h,,(3")= H L(3"(d,))i=1

Then h, is measurable for all n. Now g3,=7 if and only if 3’ is constant on the orbits of g. Butthis happens if and only if 3"(g-td)=v(d) for all de D. Therefore :1:(3")=1 if and only if f,(3"(d,))=l

698 R.A. BEKES

for all i. This shows that f(7)= imoohd7) and therefore measurable by Hewett and Stromberg

[3, 22.24b]. 13

We write D={d,d2,da } and let D.=[d dO. Let < g > be the subgroup generated by g

and < /> d the orbit of d under < g >. For each n and each k _< n let ct’(9) be the number of

distinct cycles of g such that </>dt3D, k. Form the monomial

LEMMA 2. | < (:/)eT, e. > dp{/ lira trAP(g).

PROOF. From the proof of Lemma I we saw that < ,(#)e7, e7 > h(7). So by the

dominated

if d only if 7 is coaster on the intersection of the orbits of g with D otherwise hdT)=O. Letco,rant on the intersection ofB,=

u(B. Since there e Nchoices for the vMue of on eh orbit mting D.d no restrictions

on 7 outside D., we get u(B ()...

Let Go be the subgroup of G consisting of all those g G having only a finite number of

cycles in D of length greater than 1.

LEMMA 3. I< (v) 0 yor Go.

PROOF. Suppose g ft Go. Then there either exists ko such that Cto"(g).-,oo as n.-,oo or there

exists an increasing sequence (k,,] such that ct,’() _> I. In the first case, for n>_ ko,

i=l i=l

0 _< I < (g)eT"e7 > dp(7) lin-’moot(B") < n.-.oolim N-’to’(t) O. I’n the second case we get

n- ci"(g) <_ k,,-1 and so 0 _< < ,r(g)eT, e7 > d#(7) lirmoot(B,,) -< n--,oolirn N’-(t"-’)= O.i=l So

For each k let Ft= {g G: gdi= d for all > k}. Then {FI is a nondecreasing sequence of

subgroups with Ft=Go. Suppose G {g,, g, } and let G.,= {g, g.}. Assume G isk=l

ordered in such a way that there exists a subsequence {rntj with Go c G,t= Ft.

Let F be a finite subset of G. Define the hi’cycle index of F to be the polynomial PF*1 IW’(g). Define the operator TF on X by TF TI =(g)" Write P,* for Pa," andIFgFTmfor TGm.

INFINITE VERSION OF THE POLYA ENUMERATION THEOREM 699

THEOREM 4. A6. is closed and

I < T,,,,e3‘, e7 > d(7)(k--,t an Go k-- Gn Go

PROOF. Fix k and let Dk,=(d+ 1, d+ }. If al % are representatives for the orbits

of F in SD, then Av (a, a,)xSDe. Therefore AV is closed and ,(A) . Let X be a

Hilbert space with orthonormal basis (ca: e ). By Willianson [2], s trace(TonaP, where Pr is the usual cycle index of F on D. Note that P=Pfor M1 n k. By

m, eP PF. Bym aP= aP"" ThcrefreLemma 3, G.,n Gom lira aPLemma 2, P < Te,e > d,(@. So we get o(Ar)= G Go

S

Gn GoS

Since F Go we can assume that Aao AF for all k. Therefore Aao klArt" We claim

To see this suppose that v 6 r for all k. Then there exists , 6 Aao andthat Aao=k 196 Go such that v=gv’. Since Go=U F there exists ko such that 9 Fo. Therefore v ad v,

k=1

represent the same orbit of F,oin S. Since 3‘ andT, E Ar we get 3’=7,. This proves the claim.

It follows that AGo is closed and hence measurable. Therefore U(AGo) ]moot(AV). This

completes the proof of the theorem. []

Suppose now that G is in no particular order. We show how to compute I(AGo). Let

a,,, tq Go and let Ta,.,

THEOREM 5. I(AG) m-oclim ln!m._. I < TA,,,’"e3"e3" > all,(3").S

PROOF. Exists m so that lEG,,,. Fix m>_m and let H,,, be the subgroup of GoI ifgeA,, andgenerated by A,. Define a probability measure t, on H, by u(g) A,,I

u(g) 0 otherwise. Let u be the n-fold convolution of u with itself and U the uniform

probability measure on H,,. Then by Diaconis [4, pg23], Ilu’"-UIlO where I1.11 is the

total variation lorm. If we extend the rcprcscntation , in the usual way, to the set of

measures on H, we get r(u") (Ta,,)"= Ta,,, and r(U) =Ttt,,,. It follows, therefore, that

lim < Ta,,,,,,c3’, e3" > < Ttt,,,e3", e3’ > for all "r fi St). By the dominated convergence theorem

lim [ < T e3’ > do(r) [ < TH,,,e3‘, e3" > all,(3’). Then as in the proof of Theorem 4, we/,-x:) rn’ 7,

So So

get ,(Att)= < Ttlev, e3" > dt(3"). The result follows since Go U H. 13

sDm=l

700 R.A. BEKES

3. EXAMPLE.Suppose D D, where the D are disjoint and finite and that G sends D into itself.

Then if G is G restricted to D, G is isomorphic to the product II G. In this case the

product measure on SD need no longer come from uniform measures on q.

Let S /sl,-..,sk} nd let the measure v on S be defined by u(s. ai. If IDol mdefine the measure # on S" by # H u. Let A be representatives for the orbits of G in

i=1o and PGa the cycle index. Then using the pattern inventory from P61ya’s enumerationk k oo

i=l i=l n=l

and let A be representatives for the orbits of G in RD. Then, as in the proof of Theorem 4,

! i= I,.. k and IDol = we getwe get that (A) lira ’I k()" Note that when ai ,oo=Ip(A) p, which is the situation in Theorem 4.

Now consider the ple tiled by oc unit sque tiles with sides plel to the is d

center the crdinates (, ), m d integers. We color the tiles lack or white d compute

the meure the orbits of two goups of symmetries acting on the set of such tilings. For m a

a positive integer let D {tcs ccners (, or ,: =-,-+I,...,-I,.

Let G H l: t on D: by interchging tiles with centrM crdinates (n:,k)k=l n+ln: d let H= H 1: act on D: by interchging tiles with centrM crdinates

(n:,v k=-n,..., n. Now let G H G, d H H H. With S={black, white}, wen=1 m

define probability meures g. on " by u.= H ’-, where v.(black)= , n(+l)}- +k=l

=d v.(white)= 1-v.(black). Let A(G=) =d A(H=) representatives for the orbits of G.d

H. respectively on d let A(G) d A(H) be representatives for the orbits of Gd H

resctively on S. Then .(A(H.)) ezp(-I/n9 =d so V(A(H)) _mit H .Ja(gd) > O.

n=l

REFERENCES

1. G. POLYA and R. C. READ, 17ombinatorial Enumeration of (roups, _hL and(hemic.al ompounds, Springer-Verlag, New York, 1987.

2. S. G. WILLIAMSON, Operator Theoretic Invariants and the Enumeration Theory of P61yaand de Bruijn, Journal. of (ombinatorial Theor_.y_, 1_.1 (1971), 122-138.

3. E. HEWETT and K. STROMBERG, Real and Abstract Analysis Springer-Verlag, NewYork, 1965.

4. P. DIACONIS, lepresentations in Probability and Statistics Institute ofMathematical Statistics, Hayward, California, 1988.

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